There are several possibilities.
Before I load a quilt on the frame, I measure it in six places - top border, thru the center, and bottom border for the width, and left border, thru the center, and right border for the length. These measurements should all be the same, if not then I know where the extra fullness is that I need to work in as I go. If they are not the same, perhaps you are not applying borders correctly. I have a tip sheet on applying borders on my website:
http://www.andicraftsquilting.com/learn.htm
If I'm understanding correctly, you are quilting the center of the quilt but leaving the side borders unquilted as you advance down the quilt. When I do this, I baste (either a large machine stitch or by pinning) the borders as I go. From the measurements I took earlier, I will know if I have fullness in the borders, and rather than smoothing them flat as I work down, I will distribute a little of that fullness in each pass so I can work it in evenly.
I also think you are saying that the quilt is 'hourglassing' as you are working your way down (pulling in so the center of the quilt is not as wide as the top/bottom border)? Again, fullness in the borders can cause this. If when I measured I came up with 90" top border, 88" center, 90" bottom border, I would use the 88" measurement when loading the quilt, then ease in that extra fullness. Also, are you basting the edges of the quilt before you quilt that area? I always baste the top edge and what I can reach of the sides every time I advance before I do any quilting - that helps prevent the hourglassing.
As far as what to do now, some people will take a tuck in the border, if you hand stitch it closed it will look like a seam and not be all that noticeable. Sometimes you can steam the extra fullness out of the border. If working from the front of the machine, you can use one hand to work in the fullness as you quilt. A pattern like piano keys (parallel lines an inch or two apart) can work wonders for sucking up the fullness. Or yes, you can take off and rework the borders.
Good luck, and let us know how it turns out!